Volume 524, December 2010
|Number of page(s)||23|
|Published online||22 November 2010|
Inserting the nonlinear cooling rate Eq. (6) and the injection rate into Eq. (2) gives us the differential equation for the SSC cooled electron number density nSSC (we drop the subscript in the following): (A.1)Multiplying the equation with γ2/A0 we obtain with the definitions y = A0t and S = γ2n(A.2)We yield with ξ = γ-1(A.3)If we define the implicit time variable T through (A.4)the differential equation becomes (A.5)Formally multiplying this equation with dy ! dT results in (A.6)This differential equation for the electron number density can be solved with the method of characteristics. Thus, we obtain (A.7)where ξ0 = ξ − T is a constant of integration. Equation (A.7) can be easily integrated with respect to T, which results in (A.8)We now require that S(ξ = 0,T) = 0, which means (A.9)Collecting terms, we find S to be (A.10)Since our flare begins at T = 0, we are not interested in events that take place before that moment. Hence, we find the electron number density: (A.11)We see that we have two Heaviside functions defining upper limits for γ. It is an easy task to compare them, and to find out which one is lower than the other one. Having done so, we find the solution for the nonlinearly cooled electron number density to be (A.12)yielding Eq. (7).
The time variable T has been defined through Eq. (8): (B.1) Inserting Eq. (A.10), we gain (B.2) As stated before, we are only interested in solutions for T ≥ 0. Hence, we can neglect the third Heaviside function, which results in (B.3)A first substitution w = ξ − T yields (B.4)while a second substitution w = Tv gives (B.5) where we re-substituted in the last step.
The purpose of the next two substitutions is to get rid of the time variable in the limits of the integral. In order to achieve this we first set x = γ1T and introduce g2 = γ2/γ1, which yields (B.6) Now, we use u = vx, resulting in (B.7)This integral can be expressed in terms of the hypergeometric function, but that would not yield an analytical form. Nonetheless, one can obtain an approximate solution in the regimes (small x), and x ≥ 1 (large x). An analytical continuation serves as a solution for the intermediate regime. For small x the integral can be written as (B.8) Similarly, we achieve for large x (B.9) The requirement for the solution of the intermediate regime is that it must be continuous, meaning , and U2(1) = U3(1). In order to accomplish such a behavior, we can first assume a proper solution U2 with some unspecified constants, and then try to fit it to the boundary conditions. A good ansatz is (B.10)Matching the solution with the boundary conditions, yields the values of the constants a′, b′, and c′: Since we had only two equations for three parameters, we chose a′ = 1.
Thus, the obtained solution for the intermediate x-range is (B.11)Before we summarize the results, we need to say a few words about the spectral index s. We already stated that it must be greater than 1. But according to our results above, we also find that s ≠ 3. Thus, we have two different cases to consider: s > 3, and 1 < s < 3.
Since , we can separate the variables obtaining . This can be integrated quite easily except for both intermediate cases. However, we can find approximative solutions by using the same approximations of these cases we used already during the calculation of the synchrotron spectra.
The denominator U(x) of the intermediate time regime as a function of x for three cases of s.
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The denominator U(x) (black) and its leading term (red) of the intermediate time regime as a function of x for three cases of s.
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We approximated the denominator with the leading term. The validity of this approximation for most parts of the x-range can be seen in Fig. C.2 . The largest errors occur for small values of x.
The differential Eq. (2) may look a little bit more complicated with the combined cooling term (13). However, the solution can be obtained with the methods outlined in Appendix A, yielding solution (14). The important difference is the definition of the implicit time variable, which has to be chosen as (D.1)Using the definition of U from Appendix B, this can be written as (D.2)Thus, we can use the previous results to obtain for s > 3 (D.3)and for 1 < s < 3 (D.4)where we defined , and used the leading term approximation discussed in Appendix C for the intermediate regimes.
Similarly to the steps in Appendix C, we calculate the dependence . We begin with the case s > 3. For we find (D.5)As before, we set , since . Inverting Eq. (D.5) yields (D.6)Obviously, is found from the condition yielding (D.7)For we find (D.8)or the other way around (D.9)Since is supposed to be continuous, we find the constant by matching the solutions for resulting in (D.10)We also obtain from the condition (D.11)Defining we yield for (D.12)The problem arising is that we cannot find an inverted expression for . However, we can obtain approximative results for small and large arguments of the arctan-function.
In order to achieve these approximations we define the injection parameter (D.13)for which Eq. (D.12) becomes (D.14)For α0 ≪ 1 the argument of the arctan-function is always (much) larger than unity, since . We, therefore, approximate , set , and obtain (D.15)This can be easily inverted, yielding the linear solution (D.16)Matching this solution with Eq. (D.9) yields (D.17)For α0 ≫ 1 we have to consider two cases. If , we see that . Thus, we can approximate the arctan-function to third order as , resulting in (D.18)Inverting yields (D.19)with (D.20)and (D.21)In the case , we can approximate again arctan(x/α0) ≈ π/2, yielding with (D.22)or inverted (D.23)with (D.24)What one can see here is that the injection parameter controls significantly the cooling behavior of the electrons. For α0 ≪ 1 the solution is purely linear, while for α0 ≫ 1 it is non-linear and becomes linear at later times, just as we expected.
Before we proceed with the case 1 < s < 3 we list the results of this section once more in a compact form. We also approximate the results for g2 ≫ 1, which, as one will see, simplifies a lot. with and Since in this approximation , , and, thus, one can neglect .
We will now derive the explicit form of the implicit time variable for 1 < s < 3.
The first regime is , yielding (D.37)Since , obviously d1 = 0, and the inversion becomes (D.38)with (D.39)The next time step is , resulting in (D.40)where we defined . For α0 ≪ 1 we see that (as long as s is not too close to 1 or 3), and with we approximate the integral as (D.41)The inversion is easily performed, yielding (D.42)where we obtain by matching the solutions (D.43)and (D.44)For α0 ≫ 1 we see that . As a rough approximation this is also much lower than , and, therefore, we achieve the integral (D.45)The inverted equation is (D.46)with (D.47)and (D.48)Similarly to the case for large spectral indices, we obtain here for the integral in that time regime (D.49)We will continue with the same approximations as before, yielding for α0 ≪ 1 , and with the result (D.50)The inversion is obviously (D.51)where (D.52)For α0 ≫ 1 we use for the approximation , yielding (D.53)Hence, (D.54)with (D.55)and (D.56)
The last case is for , where we can use the “linear” approximation arctan(x/α0) ≈ π/2, again. With we achieve (D.57)or inverted (D.58)where we defined (D.59)As we did for the case of large spectral indices, we sum up our results in a short list, and perform the approximation for g2 ≫ 1. with and
© ESO, 2010
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